:: Some properties of special polygonal curves
:: by Andrzej Trybulec and Yatsuka Nakamura
::
:: Received October 22, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users


theorem :: SPRECT_3:1
for D being non empty set
for f being non empty FinSequence of D
for g being FinSequence of D holds (g ^ f) /. (len (g ^ f)) = f /. (len f)
proof end;

theorem Th2: :: SPRECT_3:2
for a, b, c, d being set holds Indices ((a,b) ][ (c,d)) = {[1,1],[1,2],[2,1],[2,2]}
proof end;

theorem Th3: :: SPRECT_3:3
for n being Nat
for p, q being Point of (TOP-REAL n)
for r being Real st 0 < r & p = ((1 - r) * p) + (r * q) holds
p = q
proof end;

theorem Th4: :: SPRECT_3:4
for n being Nat
for p, q being Point of (TOP-REAL n)
for r being Real st r < 1 & p = ((1 - r) * q) + (r * p) holds
p = q
proof end;

theorem :: SPRECT_3:5
for n being Nat
for p, q being Point of (TOP-REAL n) st p = (1 / 2) * (p + q) holds
p = q
proof end;

theorem Th6: :: SPRECT_3:6
for n being Nat
for p, q, r being Point of (TOP-REAL n) st q in LSeg (p,r) & r in LSeg (p,q) holds
q = r
proof end;

theorem Th7: :: SPRECT_3:7
for A being non empty Subset of (TOP-REAL 2)
for p being Element of (Euclid 2)
for r being Real st A = Ball (p,r) holds
A is connected
proof end;

theorem Th8: :: SPRECT_3:8
for A, B being Subset of (TOP-REAL 2) st A is open & B is_a_component_of A holds
B is open
proof end;

theorem :: SPRECT_3:9
for p, q, r, s being Point of (TOP-REAL 2) st LSeg (p,q) is horizontal & LSeg (r,s) is horizontal & LSeg (p,q) meets LSeg (r,s) holds
p `2 = r `2
proof end;

theorem :: SPRECT_3:10
for p, q, r being Point of (TOP-REAL 2) st LSeg (p,q) is vertical & LSeg (q,r) is horizontal holds
(LSeg (p,q)) /\ (LSeg (q,r)) = {q}
proof end;

theorem :: SPRECT_3:11
for p, q, r, s being Point of (TOP-REAL 2) st LSeg (p,q) is horizontal & LSeg (s,r) is vertical & r in LSeg (p,q) holds
(LSeg (p,q)) /\ (LSeg (s,r)) = {r}
proof end;

theorem :: SPRECT_3:12
for i, j, k being Nat
for G being Go-board st 1 <= j & j <= k & k <= width G & 1 <= i & i <= len G holds
(G * (i,j)) `2 <= (G * (i,k)) `2
proof end;

theorem :: SPRECT_3:13
for i, j, k being Nat
for G being Go-board st 1 <= j & j <= width G & 1 <= i & i <= k & k <= len G holds
(G * (i,j)) `1 <= (G * (k,j)) `1
proof end;

theorem Th14: :: SPRECT_3:14
for C being Subset of (TOP-REAL 2) holds LSeg ((NW-corner C),(NE-corner C)) c= L~ (SpStSeq C)
proof end;

theorem Th15: :: SPRECT_3:15
for C being non empty compact Subset of (TOP-REAL 2) holds N-min C in LSeg ((NW-corner C),(NE-corner C))
proof end;

registration
let C be Subset of (TOP-REAL 2);
cluster LSeg ((NW-corner C),(NE-corner C)) -> horizontal ;
coherence
LSeg ((NW-corner C),(NE-corner C)) is horizontal
proof end;
end;

theorem :: SPRECT_3:16
for g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st g /. 1 <> p & ( (g /. 1) `1 = p `1 or (g /. 1) `2 = p `2 ) & g is being_S-Seq & (LSeg (p,(g /. 1))) /\ (L~ g) = {(g /. 1)} holds
<*p*> ^ g is being_S-Seq
proof end;

theorem :: SPRECT_3:17
for j being Nat
for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2) st 1 < j & j <= len f & p in L~ (mid (f,1,j)) holds
LE p,f /. j, L~ f,f /. 1,f /. (len f)
proof end;

theorem :: SPRECT_3:18
for i, j being Nat
for h being FinSequence of (TOP-REAL 2) st i in dom h & j in dom h holds
L~ (mid (h,i,j)) c= L~ h
proof end;

theorem :: SPRECT_3:19
for i, j being Nat st 1 <= i & i < j holds
for f being FinSequence of (TOP-REAL 2) st j <= len f holds
L~ (mid (f,i,j)) = (LSeg (f,i)) \/ (L~ (mid (f,(i + 1),j)))
proof end;

theorem :: SPRECT_3:20
for i, j being Nat
for f being FinSequence of (TOP-REAL 2) st 1 <= i & i < j & j <= len f holds
L~ (mid (f,i,j)) = (L~ (mid (f,i,(j -' 1)))) \/ (LSeg (f,(j -' 1)))
proof end;

theorem :: SPRECT_3:21
for f, g being FinSequence of (TOP-REAL 2) st f is being_S-Seq & g is being_S-Seq & ( (f /. (len f)) `1 = (g /. 1) `1 or (f /. (len f)) `2 = (g /. 1) `2 ) & L~ f misses L~ g & (LSeg ((f /. (len f)),(g /. 1))) /\ (L~ f) = {(f /. (len f))} & (LSeg ((f /. (len f)),(g /. 1))) /\ (L~ g) = {(g /. 1)} holds
f ^ g is being_S-Seq
proof end;

theorem :: SPRECT_3:22
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
(R_Cut (f,p)) /. 1 = f /. 1
proof end;

theorem :: SPRECT_3:23
for j being Nat
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg ((f /. j),p) holds
LE q,p, L~ f,f /. 1,f /. (len f)
proof end;

theorem Th24: :: SPRECT_3:24
for f being V22() standard special_circular_sequence holds
( LeftComp f is open & RightComp f is open )
proof end;

registration
let f be V22() standard special_circular_sequence;
cluster L~ f -> non horizontal non vertical ;
coherence
( not L~ f is vertical & not L~ f is horizontal )
proof end;
cluster LeftComp f -> being_Region ;
coherence
LeftComp f is being_Region
proof end;
cluster RightComp f -> being_Region ;
coherence
RightComp f is being_Region
proof end;
end;

theorem Th25: :: SPRECT_3:25
for f being V22() standard special_circular_sequence holds RightComp f misses L~ f
proof end;

theorem Th26: :: SPRECT_3:26
for f being V22() standard special_circular_sequence holds LeftComp f misses L~ f
proof end;

theorem Th27: :: SPRECT_3:27
for f being V22() standard special_circular_sequence holds i_w_n f < i_e_n f
proof end;

theorem Th28: :: SPRECT_3:28
for f being V22() standard special_circular_sequence ex i being Nat st
( 1 <= i & i < len (GoB f) & N-min (L~ f) = (GoB f) * (i,(width (GoB f))) )
proof end;

theorem Th29: :: SPRECT_3:29
for i being Nat
for f being V22() standard clockwise_oriented special_circular_sequence st i in dom (GoB f) & f /. 1 = (GoB f) * (i,(width (GoB f))) & f /. 1 = N-min (L~ f) holds
( f /. 2 = (GoB f) * ((i + 1),(width (GoB f))) & f /. ((len f) -' 1) = (GoB f) * (i,((width (GoB f)) -' 1)) )
proof end;

theorem :: SPRECT_3:30
for i, j being Nat
for f being V22() standard special_circular_sequence st 1 <= i & i < j & j <= len f & f /. 1 in L~ (mid (f,i,j)) & not i = 1 holds
j = len f
proof end;

theorem :: SPRECT_3:31
for f being V22() standard clockwise_oriented special_circular_sequence st f /. 1 = N-min (L~ f) holds
LSeg ((f /. 1),(f /. 2)) c= L~ (SpStSeq (L~ f))
proof end;

theorem Th32: :: SPRECT_3:32
for f being rectangular FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) holds
( not p in L~ f or p `1 = W-bound (L~ f) or p `1 = E-bound (L~ f) or p `2 = S-bound (L~ f) or p `2 = N-bound (L~ f) )
proof end;

registration
cluster V1() V4( NAT ) V5( the U1 of (TOP-REAL 2)) non empty Function-like V28() FinSequence-like FinSubsequence-like V241( the U1 of (TOP-REAL 2)) special unfolded s.c.c. rectangular for FinSequence of the U1 of (TOP-REAL 2);
existence
ex b1 being special_circular_sequence st b1 is rectangular
proof end;
end;

theorem Th33: :: SPRECT_3:33
for f being rectangular special_circular_sequence
for g being S-Sequence_in_R2 st g /. 1 in LeftComp f & g /. (len g) in RightComp f holds
L~ f meets L~ g
proof end;

theorem Th34: :: SPRECT_3:34
for f being rectangular special_circular_sequence holds SpStSeq (L~ f) = f
proof end;

theorem Th35: :: SPRECT_3:35
for f being rectangular special_circular_sequence holds L~ f = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = W-bound (L~ f) & p `2 <= N-bound (L~ f) & p `2 >= S-bound (L~ f) ) or ( p `1 <= E-bound (L~ f) & p `1 >= W-bound (L~ f) & p `2 = N-bound (L~ f) ) or ( p `1 <= E-bound (L~ f) & p `1 >= W-bound (L~ f) & p `2 = S-bound (L~ f) ) or ( p `1 = E-bound (L~ f) & p `2 <= N-bound (L~ f) & p `2 >= S-bound (L~ f) ) ) }
proof end;

theorem Th36: :: SPRECT_3:36
for f being rectangular special_circular_sequence holds GoB f = ((f /. 4),(f /. 1)) ][ ((f /. 3),(f /. 2))
proof end;

theorem Th37: :: SPRECT_3:37
for f being rectangular special_circular_sequence holds
( LeftComp f = { p where p is Point of (TOP-REAL 2) : ( not W-bound (L~ f) <= p `1 or not p `1 <= E-bound (L~ f) or not S-bound (L~ f) <= p `2 or not p `2 <= N-bound (L~ f) ) } & RightComp f = { q where q is Point of (TOP-REAL 2) : ( W-bound (L~ f) < q `1 & q `1 < E-bound (L~ f) & S-bound (L~ f) < q `2 & q `2 < N-bound (L~ f) ) } )
proof end;

registration
cluster V1() V4( NAT ) V5( the U1 of (TOP-REAL 2)) non empty non trivial V16() Function-like non constant V28() FinSequence-like FinSubsequence-like V241( the U1 of (TOP-REAL 2)) special unfolded s.c.c. standard rectangular clockwise_oriented for FinSequence of the U1 of (TOP-REAL 2);
existence
ex b1 being rectangular special_circular_sequence st b1 is clockwise_oriented
proof end;
end;

registration
cluster non empty V241( the U1 of (TOP-REAL 2)) special unfolded s.c.c. rectangular -> rectangular clockwise_oriented for FinSequence of the U1 of (TOP-REAL 2);
coherence
for b1 being rectangular special_circular_sequence holds b1 is clockwise_oriented
proof end;
end;

theorem :: SPRECT_3:38
for f being rectangular special_circular_sequence
for g being S-Sequence_in_R2 st g /. 1 in LeftComp f & g /. (len g) in RightComp f holds
Last_Point ((L~ g),(g /. 1),(g /. (len g)),(L~ f)) <> NW-corner (L~ f)
proof end;

theorem :: SPRECT_3:39
for f being rectangular special_circular_sequence
for g being S-Sequence_in_R2 st g /. 1 in LeftComp f & g /. (len g) in RightComp f holds
Last_Point ((L~ g),(g /. 1),(g /. (len g)),(L~ f)) <> SE-corner (L~ f)
proof end;

theorem Th40: :: SPRECT_3:40
for f being rectangular special_circular_sequence
for p being Point of (TOP-REAL 2) st ( W-bound (L~ f) > p `1 or p `1 > E-bound (L~ f) or S-bound (L~ f) > p `2 or p `2 > N-bound (L~ f) ) holds
p in LeftComp f
proof end;

theorem :: SPRECT_3:41
for f being V22() standard clockwise_oriented special_circular_sequence st f /. 1 = N-min (L~ f) holds
LeftComp (SpStSeq (L~ f)) c= LeftComp f
proof end;

theorem Th42: :: SPRECT_3:42
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) holds
( <*p,q*> is_in_the_area_of f iff ( <*p*> is_in_the_area_of f & <*q*> is_in_the_area_of f ) )
proof end;

theorem Th43: :: SPRECT_3:43
for f being rectangular FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st <*p*> is_in_the_area_of f & ( p `1 = W-bound (L~ f) or p `1 = E-bound (L~ f) or p `2 = S-bound (L~ f) or p `2 = N-bound (L~ f) ) holds
p in L~ f
proof end;

theorem Th44: :: SPRECT_3:44
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2)
for r being Real st 0 <= r & r <= 1 & <*p,q*> is_in_the_area_of f holds
<*(((1 - r) * p) + (r * q))*> is_in_the_area_of f
proof end;

theorem Th45: :: SPRECT_3:45
for i being Nat
for f, g being FinSequence of (TOP-REAL 2) st g is_in_the_area_of f & i in dom g holds
<*(g /. i)*> is_in_the_area_of f
proof end;

theorem Th46: :: SPRECT_3:46
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st g is_in_the_area_of f & p in L~ g holds
<*p*> is_in_the_area_of f
proof end;

theorem Th47: :: SPRECT_3:47
for f being rectangular FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st not q in L~ f & <*p,q*> is_in_the_area_of f holds
(LSeg (p,q)) /\ (L~ f) c= {p}
proof end;

theorem :: SPRECT_3:48
for f being rectangular FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & not q in L~ f & <*q*> is_in_the_area_of f holds
(LSeg (p,q)) /\ (L~ f) = {p}
proof end;

theorem Th49: :: SPRECT_3:49
for i, j being Nat
for f being V22() standard special_circular_sequence st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
<*((GoB f) * (i,j))*> is_in_the_area_of f
proof end;

theorem :: SPRECT_3:50
for g being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st <*p,q*> is_in_the_area_of g holds
<*((1 / 2) * (p + q))*> is_in_the_area_of g
proof end;

theorem :: SPRECT_3:51
for f, g being FinSequence of (TOP-REAL 2) st g is_in_the_area_of f holds
Rev g is_in_the_area_of f
proof end;

theorem :: SPRECT_3:52
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st g is_in_the_area_of f & <*p*> is_in_the_area_of f & g is being_S-Seq & p in L~ g holds
R_Cut (g,p) is_in_the_area_of f
proof end;

theorem :: SPRECT_3:53
for f being V22() standard special_circular_sequence
for g being FinSequence of (TOP-REAL 2) holds
( g is_in_the_area_of f iff g is_in_the_area_of SpStSeq (L~ f) )
proof end;

theorem :: SPRECT_3:54
for f being rectangular special_circular_sequence
for g being S-Sequence_in_R2 st g /. 1 in LeftComp f & g /. (len g) in RightComp f holds
L_Cut (g,(Last_Point ((L~ g),(g /. 1),(g /. (len g)),(L~ f)))) is_in_the_area_of f
proof end;

theorem :: SPRECT_3:55
for i, j being Nat
for f being V22() standard special_circular_sequence st 1 <= i & i < len (GoB f) & 1 <= j & j < width (GoB f) holds
Int (cell ((GoB f),i,j)) misses L~ (SpStSeq (L~ f))
proof end;

theorem :: SPRECT_3:56
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st g is_in_the_area_of f & <*p*> is_in_the_area_of f & g is being_S-Seq & p in L~ g holds
L_Cut (g,p) is_in_the_area_of f
proof end;